midterm_1_review (dragged) 1

midterm_1_review (dragged) 1 - 2 MA 36600 MIDTERM#1 REVIEW...

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Unformatted text preview: 2 MA 36600 MIDTERM #1 REVIEW §1.3: Classification of Differential Equations. • There are two types of differential equations: – ODEs or ordinary differential equations are equations which do not involve partial derivatives. – PDEs or partial differential equations are equations which do involve partial derivatives. • Consider a function y = y (t), and let y (k) denote the k th derivative. An nth order differential equation is an ordinary differential equation in the form ￿ ￿ ￿ ￿ F t, y, y (1) , y (2) , . . . , y (n) = 0 =⇒ y (n) = G t, y, y (1) , y (2) , . . . , y (n−1) . • We say that this differential equation is linear if G(t, ￿ ) = g0 (t) + g1 (t) y1 + g2 (t) y2 + · · · + gn (t) yn y is a linear function for the vector ￿ = (y1 , y2 , . . . , yn ). Otherwise, we say that this differential y equation is nonlinear. Chapter 2 §2.1: Linear Equations; Method of Integrating Factors. • An ordinary differential equation is said to be a first order equation if it is a relation in the form F (t, y, y ￿ ) = 0 =⇒ dy = G(t, y ). dt We say that this first order equation is linear if G(t, y ) = g (t) − p(t) y =⇒ dy + p(t) y = g (t). dt • Say that we can find a function µ = µ(t) such that ￿ dy d￿ + µ(t) p(t) y = µ(t) y . dt dt Such a function is called an integrating factor. More precisely, it is a function such that i. µ(t) is not identically zero, and ii. µ￿ (t) = p(t) µ(t). • The general solution of y ￿ + p(t) y = g (t) is the function ￿￿ t ￿ ￿￿ t ￿ 1 µ(τ ) g (τ ) dτ + C where µ(t) = exp p(τ ) dτ . y (t) = µ(t) µ(t) §2.2: Separable Equations. • Consider an ordinary differential equation in the form y ￿ = G(t, y ). We say that it is separable if we have a factorization G(t, y ) = Y (y ) T (t). • To solve such an equation, the trick is to bring all of the terms involving y to one side, and all of the terms involving t to the other: ￿ ￿ dy 1 dy 1 = Y (y ) T (t) =⇒ = T (t) =⇒ dy = T (t) dt + C. dt Y (y ) dt Y (y ) The general solution is f (t, y ) = C where f (t, y ) = f1 (y ) − f2 (t) in terms of the antiderivatives ￿y ￿t 1 f1 (y ) = dσ and f2 (t) = T (τ ) dτ . Y (σ ) • The expression “f (t, y ) = C ” is an implicit solution to the differential equation y ￿ = G(t, y ). The constant C can be found from the initial conditions. The graphs of the expressions f (t, y ) = C in the ty -plane are called integral curves. ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.

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